A NOTE ON THE SYMPLECTIC STRUCTURE ON THE SPACE OF G-MONOPOLES

MICHAEL FINKELBERG, ALEXANDER KUZNETSOV, AND IVAN MIRKOVIĆ

1. Introduction

1.1. Let G be a semisimple complex Lie group with the Cartan datum (I, ·) and the root datum (Y, X, …). Let H ⊂ B = B_{+}, B_{−} ⊂ G be a Cartan subgroup and a pair of opposite Borel subgroups respectively. Let X = G/B be the flag manifold of G. Let C = ℙ^{1} ∋ ∞ be the projective line. Let α = ∑_{i∈I} a_{i}i ∈ ℕ[I] = H_{2}(X, ℤ).

The moduli space of G-monopoles of topological charge α (see e.g. [4]) is naturally identified with the space M_{b}(X, α) of based maps from (C, ∞) to (X, B_{+}). The moduli space of G-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. We propose a simple explicit formula for the symplectic structure on M_{b}(X, α). It generalizes the well known formula for G = SL_{2} [1].

1.2. Recall that for G = SL_{2} we have (X, B_{+}) = (ℙ^{1}, ∞). Recall the natural local coordinates on M_{b}(ℙ^{1}, a) (see [1]). We fix a coordinate z on C such that z(∞) = ∞.

Then a based map ϕ : (C, ∞) → (ℙ^{1}, ∞) of degree a is a rational function \frac{p(z)}{q(z)} where

p(z) is a degree a polynomial with the leading coefficient 1, and q(z) is a degree < a polynomial. Let U be the open subset of based maps such that the roots x^{1}, …, x^{a} of p(z) are multiplicity free. Let y^{k} be the value of q(z) at x^{k}. Then x^{1}, …, x^{a}, y^{1}, …, y^{a} form an étale coordinate system on U. The symplectic form Ω on M_{b}(ℙ^{1}, a) equals

∑_{k=1}^{a} \frac{dy^{k} ∧ dx^{k}}{y^{k}}. In other words, the Poisson brackets of these coordinates are as follows:

{x^{k}, x^{m}} = 0 = {y^{k}, y^{m}}; {x^{k}, y^{m}} = δ_{km}y^{m}.

For an arbitrary G and i ∈ I let X_{i} ⊂ X be the corresponding codimension 1 B_{−}-orbit (Schubert cell), and let \overline{X}_{i} ⊃ X_{i} be its closure (Schubert variety). For ϕ ∈ M_{b}(X, α) we define x_{i}^{1}, …, x_{i}^{a_{i}} ∈ 𝔸^{1} as the points of intersection of ϕ(ℙ^{1}) with \overline{X}_{i}. This way we obtain the projection π^{α} : M_{b}(X, α) → 𝔸^{α} (the configuration space of I-colored divizors of degree α on 𝔸^{1}). Let U ⊂ M_{b}(X, α) be the open subset of based maps such that ϕ(ℙ^{1}) ∩ \overline{X}_{i} ⊂ X_{i} for any i ∈ I, and x_{i}^{k} ≠ x_{j}^{l} for any i, j ∈ I, 1 ≤ k ≤ a_{i}, 1 ≤ l ≤ a_{j}. Locally in X the cell X_{i} is the zero divizor of a function φ_{i} (globally, φ_{i} is a section of the line bundle L_{ω_{i}} corresponding to the fundamental weight ω_{i} ∈ X). The rational function φ_{i} ∘ ϕ on C is of the form \frac{p_{i}(z)}{q_{i}(z)} where p_{i}(z) is a degree a_{i} polynomial with the leading

Date: March 1998.

M.F. and A.K. were partially supported by CRDF grant RM1-265.

coefficient 1, and q_{i}(z) is a degree < a_{i} polynomial. Let y_{i}^{k} be the value of q_{i}(z) at x_{i}^{k}. Then x_{i}^{k}, y_{i}^{k}, i ∈ I, 1 ≤ k ≤ a_{i}, form an étale coordinate system on U. The Poisson brackets of these coordinates are as follows:

\{x_{i}^{k}, x_{j}^{l}\} = 0 = \{y_{i}^{k}, y_{i}^{l}\}; \{x_{i}^{k}, y_{j}^{l}\} = δ_{ij}δ_{kl}y_{j}^{l}; \{y_{i}^{k}, y_{j}^{l}\} = i · j \frac{y_{i}^{k}y_{j}^{l}}{x_{i}^{k} − x_{j}^{l}} for i ≠ j.

1.3. It follows that the symmetric functions of the x-coordinates (well defined on the whole M_{b}(X, α)) are in involution. In other words, the projection π^{α} : M_{b}(X, α) → 𝔸^{α} is an integrable system on M_{b}(X, α). The fibers of π^{α} : U → 𝔸^{α} are Lagrangian submanifolds of U. It is known that all the fibers of π^{α} : M_{b}(X, α) → 𝔸^{α} are equidimensional of the same dimension |α| (see [3]), hence π^{α} is flat, hence all the fibers are Lagrangian.

1.4. Let P ⊃ B be a parabolic subgroup. The construction of the Poisson structure on M_{b}(X, α) generalizes verbatim to the space of based maps M = M_{b}(G/P, β). In most cases the corresponding map T M → T^{∗}M is not an isomorphism, i.e. M splits into nontrivial symplectic leaves.

1.5. The above Poisson structure is a baby (rational) version of the Poisson structure on the moduli space of B-bundles over an elliptic curve [2]. We learnt of its definition (as a differential in the hypercohomology spectral sequence, see §2) from N.Markaryan. Thus, our modest contribution reduces just to a proof of Jacobi identity. Note that the Poisson structure of [2] arises as a quasiclassical limit of elliptic algebras. On the other hand, M_{b}(X, α) is an open subset in the moduli space M̂_{b}(X, α) of B-bundles on C trivialized at ∞, such that the induced H-bundle has degree α. One can see easily that M̂_{b}(X, α) is isomorphic to an affine space 𝔸^{2|α|}, and the symplectic structure on M_{b}(X, α) extends to the Poisson structure on M̂_{b}(X, α). The latter one can be quantized along the lines of [2].

It is clear from the above discussion that the present note owes its existense to the generous explanations of B.Feigin and N.Markaryan.

1.6. Notations. For a subset J ⊂ I we denote by P_{J} ⊃ B the corresponding parabolic subgroup. Thus, P_{∅} = B. Denote by X_{J} = G/P_{J} the corresponding parabolic flag variety; thus, X_{∅} = X. We denote by ϖ : X → X_{J} the natural projection. We denote by x ∈ X_{J} the marked point ϖ(B_{+}).

Let M = M_{b}(X_{J}, α) denote the space of based algebraic maps ϕ : (C, ∞) → (X_{J}, x) of degree α ∈ H_{2}(X_{J}, ℤ).

Let 𝔤 denote the Lie algebra of G. Let 𝔤_{X_{J}} denote the trivial vector bundle with the fiber 𝔤 over X_{J} and let 𝔭_{X_{J}} ⊂ 𝔤_{X_{J}} (resp. 𝔯_{X_{J}} ⊂ 𝔭_{X_{J}} ⊂ 𝔤_{X_{J}}) be its subbundle with the fiber over a point P equal to the corresponding Lie subalgebra 𝔭 ⊂ 𝔤 (resp. its nilpotent radical 𝔯 ⊂ 𝔭 ⊂ 𝔤). In case J = ∅ we will also denote 𝔯_{X_{∅}} by 𝔫_{X}, and 𝔭_{X_{∅}} by 𝔟_{X}. Note that the quotient bundle 𝔥_{X} := 𝔟_{X}/𝔫_{X} is trivial (abstract Cartan algebra).

Recall that the tangent bundle T X_{J} of X_{J} (resp. cotangent bundle T^{∗}X_{J}) is canonicaly isomorphic to the bundle 𝔤_{X_{J}} /𝔭_{X_{J}} (resp. 𝔯_{X_{J}}).

1.7. Acknowledgments. This paper has been written during the stay of the second author at the Max-Planck-Institut für Mathematik. He would like to express his sincere gratitude to the Institut for the hospitality and the excellent work conditions.

2. The Poisson structure

2.1. The fibers of the tangent and cotangent bundles of the space M at the point ϕ are computed as follows:

\begin{aligned}T_{ϕ}M & = H^{0}(C, (ϕ^{∗}T X_{J}) ⊗ O_{C}(−1)) & = H^{0}(C, (ϕ^{∗}𝔤_{X_{J}} /𝔭_{X_{J}}) ⊗ O_{C}(−1)), \\ T_{ϕ}^{∗}M & = H^{1}(C, (ϕ^{∗}T^{∗}X_{J}) ⊗ O_{C}(−1)) & = H^{1}(C, (ϕ^{∗}𝔯_{X_{J}}) ⊗ O_{C}(−1)). \end{aligned}

The second identification follows from the first by the Serre duality.

We have a tautological complex of vector bundles on X_{J}:

𝔯_{X_{J}} → 𝔤_{X_{J}} → 𝔤_{X_{J}} /𝔭_{X_{J}}  (1)

The pull-back via ϕ of this complex twisted by O_{C}(−1) gives the following complex of vector bundles on C

(ϕ^{∗}𝔯_{X_{J}}) ⊗ O_{C}(−1) → (ϕ^{∗}𝔤_{X_{J}}) ⊗ O_{C}(−1) → (ϕ^{∗}𝔤_{X_{J}} /𝔭_{X_{J}}) ⊗ O_{C}(−1)  (2)

Consider the hypercohomology spectral sequence of the complex (2). Since 𝔤_{X_{J}} is the trivial vector bundle we have H^{•}(C, (ϕ^{∗}𝔤_{X_{J}}) ⊗ O_{C}(−1)) = 0, hence the second differential of the spectral sequence induces a map

d_{2} : H^{1}(C, (ϕ^{∗}𝔯_{X_{J}}) ⊗ O_{C}(−1)) → H^{0}(C, (ϕ^{∗}𝔤_{X_{J}} /𝔭_{X_{J}}) ⊗ O_{C}(−1))

that is a map P_{ϕ}^{X_{J}} : T_{ϕ}^{∗}M → T_{ϕ}M. This construction easily globalizes to give a morphism P^{X_{J}} : T^{∗}M → T M.

Theorem 1. P defines a Poisson structure on M.

Here we will reduce the Theorem to the case X_{J} = X. This case will be treated in the next section.

2.2. Let ϖ_{∗} : H_{2}(X, ℤ) → H_{2}(X_{J}, ℤ) be the push-forward map. The map ϖ induces a map Π : M_{b}(X, α) → M_{b}(X_{J}, ϖ_{∗}α).

Proposition 1. The map Π respects P, that is the following square is commutative

T^{∗}M_{b}(X, α) ⟶^{P^{X}} T M_{b}(X, α)
Π^{∗}↑  ^{Π_{∗}}↓
Π^{∗}T^{∗}M_{b}(X_{J}, ϖ_{∗}α) ⟶^{P^{X_{J}}} Π^{∗}T M_{b}(X_{J}, ϖ_{∗}α)

Proof. We have the following commutative square on X

𝔫_{X} ⟶ 𝔤_{X} ⟶ 𝔤_{X}/𝔟_{X}
↑  ∥  ∥
ϖ^{∗}𝔯_{X_{J}} ⟶ 𝔤_{X} ⟶ 𝔤_{X}/𝔟_{X}
∥  ∥  ↓
ϖ^{∗}𝔯_{X_{J}} ⟶ ϖ^{∗}𝔤_{X_{J}} ⟶ ϖ^{∗}(𝔤_{X_{J}} /𝔭_{X_{J}})

Consider its pull-back via ϕ ∈ M_{b}(X, α) twisted by O_{C}(−1). Let d_{2} denote the second differential of the hypercohomology spectral sequence of the middle row. Then we evidently have

P^{X} · Π^{∗} = d_{2}, Π_{∗} · d_{2} = P^{X_{J}}

and the Proposition follows.

Now, assume that we have proved that P^{X} defines a Poisson structure. For any β ∈ H_{2}(X_{J}, ℤ) we can choose α ∈ H_{2}(X, ℤ) such that ϖ_{∗}α = β and the map Π is open. Then the algebra of functions on M_{b}(X_{J}, β) is embedded into the algebra of functions on M_{b}(X, α) and the Proposition 1 shows that the former bracket is induced by the latter one. Hence it is also a Poisson bracket.

3. The case of X

In this section we will denote M_{b}(X, α) simply by M.

3.1. Since 𝔥 is a trivial vector bundle on X the exact sequences

0 → 𝔫_{X} → 𝔟_{X} → 𝔥_{X} → 0, 0 → 𝔥_{X} → 𝔤_{X}/𝔫_{X} → 𝔤_{X}/𝔟_{X} → 0

induce the isomorphisms

\begin{aligned}T_{ϕ}M & = H^{0}(C, (ϕ^{∗}𝔤_{X}/𝔟_{X}) ⊗ O_{C}(−1)) & = H^{0}(C, (ϕ^{∗}𝔤_{X}/𝔫_{X}) ⊗ O_{C}(−1)), \\ T_{ϕ}^{∗}M & = H^{1}(C, (ϕ^{∗}𝔫_{X}) ⊗ O_{C}(−1)) & = H^{1}(C, (ϕ^{∗}𝔟_{X}) ⊗ O_{C}(−1)) \end{aligned}

Applying the construction of 2.1 to the following tautological complex of vector bundles on X

𝔟_{X} → 𝔤_{X} ⊕ 𝔥_{X} → 𝔤_{X}/𝔫_{X}  (3)

and taking into account the above isomorphisms we get a map P̃_{ϕ}^{X} : T_{ϕ}^{∗}M → T_{ϕ}M.

Lemma 1. We have P̃_{ϕ}^{X} = P_{ϕ}^{X}.

Proof. The same reasons as in the proof of the Proposition 1 work if we consider the following commutative diagram

𝔫_{X} ⟶ 𝔤_{X} ⟶ 𝔤_{X}/𝔟_{X}
∥  ∥  ↑
𝔫_{X} ⟶ 𝔤_{X} ⟶ 𝔤_{X}/𝔫_{X}
↓  ↓  ∥
𝔟_{X} ⟶ 𝔤_{X} ⊕ 𝔥_{X} ⟶ 𝔤_{X}/𝔫_{X}

It will be convenient for us to use the complex (3) for the definition of the map P_{ϕ}^{X} instead of (1).

3.2. Here we will describe the Plücker embedding of the space M.

Let X ⊃ ℧ ≅ I be the set of fundamental weights: ⟨i, ω_{j}⟩ = δ_{ij}. We denote by (,) the scalar product on X such that (i′, j′) = i · j for the simple roots i′, j′. For a dominant weight λ ∈ X we denote by V_{λ} the irreducible G-module with highest weight λ.

Recall that X is canonically embedded into the product of projective spaces

X ⊂ ∏_{ω∈℧}ℙ(V_{ω})

This induces the embedding

M ⊂ ∏_{ω∈℧}M_{b}(ℙ(V_{ω}), ⟨α, ω⟩)

Note that the marked point of the space ℙ(V_{ω}) is just the highest weight vector v_{ω} with respect to the Borel subgroup B.

A degree d based map ϕ_{ω} : (C, ∞) → (ℙ(V_{ω}), v_{ω}) can be represented by a V_{ω}-valued degree d polynomial in z, taking the value v_{ω} at infinity. Let us denote the affine space of such polynomials by R_{d}(V_{ω}).

The Plücker embedding of the space M is the embedding into the product of affine spaces

M ⊂ ∏_{ω∈℧}R_{⟨α,ω⟩}(V_{ω}).

A map ϕ ∈ M will be represented by a collection of polynomials (ϕ_{ω} ∈ R_{⟨α,ω⟩}(V_{ω}))_{ω∈℧}.

3.3. The coordinates. The dual representation V_{ω}^{∗} decomposes into the sum of weight subspaces

V_{ω}^{∗} = ⊕_{λ∈X}V^{∗}_{ω}^{λ}.

We choose a weight base (f_{ω}^{λ}) of V_{ω}^{∗}, such that f_{ω}^{−ω}(v_{ω}) = 1. Suppose ⟨i, ω⟩ = 1. Then dim V^{∗}_{ω}^{−ω} = dim V^{∗}_{ω}^{−ω+i′} = 1, and dim V^{∗}_{ω}^{−ω+i′+j′} = 0 if i · j = 0, and dim V^{∗}_{ω}^{−ω+i′+j′} = 1 if i · j ≠ 0. Hence, in the latter case, the vectors f_{ω}^{−ω}, f_{ω}^{i′−ω} and f_{ω}^{i′+j′−ω} are defined uniquely up to multiplication by a constant. Let E_{i}, F_{i}, H_{i} be the standard generators of 𝔤. Then we will take f_{ω}^{i′−ω} := E_{i}f_{ω}^{−ω}, f_{ω}^{i′+j′−ω} := E_{j}E_{i}f_{ω}^{−ω}.

We consider the polynomials ϕ_{ω}^{λ} := f_{ω}^{λ}(ϕ_{ω}): the λ weight components of ϕ_{ω}. In particular, ϕ_{ω}^{−ω} is the degree ⟨α, ω⟩ unitary polynomial and ϕ_{ω}^{i′−ω} is the degree < ⟨α, ω⟩ polynomial.

Let x_{ω}^{1}, …, x_{ω}^{⟨α,ω⟩} be the roots of ϕ_{ω}^{−ω} and y_{ω}^{1}, …, y_{ω}^{⟨α,ω⟩} be the values of ϕ_{ω}^{i′−ω} at the points x_{ω}^{1}, …, x_{ω}^{⟨α,ω⟩} respectively. Consider the open subset U ⊂ M formed by all the maps ϕ such that all x_{ω}^{k} are distinct and all y_{ω}^{k} are non-zero. On this open set we have

ϕ_{ω}^{−ω}(z) = ∏_{k=1}^{⟨α,ω⟩}(z − x_{ω}^{k}), ϕ_{ω}^{i′−ω}(z) = ∑_{k=1}^{⟨α,ω⟩} \frac{y_{ω}^{k}ϕ_{ω}^{−ω}(z)}{(ϕ_{ω}^{−ω})′(x_{ω}^{k})(z − x_{ω}^{k})}.

The collection of 2|α| functions

(x_{ω}^{k}, y_{ω}^{k}),  (ω ∈ ℧, 1 ≤ k ≤ ⟨α, ω⟩)  (4)

is an étale coordinate system in U. One can either check this straightforwardly, or just note that the matrix of P^{X} in these coordinates has a maximal rank, see the Remark 2 below.

So let us compute the map P^{X} in these coordinates.

3.4. The action of 𝔤 on V_{ω} induces an embedding

𝔤_{X}/𝔫_{X} ⊂ ⊕_{ω∈℧}V_{ω} ⊗ L_{ω}

of vector bundles over X and the dual surjection

⊕_{ω∈℧}V_{ω}^{∗} ⊗ L_{ω}^{∗} → 𝔟_{X},

where L_{ω} stands for the line bundle, corresponding to the weight ω. Hence we have the following complex

⊕_{ω∈℧}V_{ω}^{∗} ⊗ L_{ω}^{∗} → 𝔤_{X} ⊕ 𝔥_{X} → ⊕_{ω∈℧}V_{ω} ⊗ L_{ω}  (5)

Remark 1. The differentials in the above complex in the fiber over a point B′ ∈ X are computed as follows:

φ ∈ V_{ω}^{∗} ↦ (∑ φ(ξ^{k}v′)ξ_{k}) ⊕ (∑ ω(h^{i})φ(v′)h_{i}) ∈ 𝔤 ⊕ 𝔥,
ξ ⊕ h ∈ 𝔤 ⊕ 𝔥 ↦ ξv′ − ω(h)v′ ∈ V_{ω}.

Here v′ is a highest weight vector of V_{ω} with respect to B′; (ξ_{k}), (ξ^{k}) are dual (with respect to the standard scalar product) bases of 𝔤; and (h_{i}), (h^{i}) are dual bases of 𝔥.

3.5. In order to compute the brackets of the coordinates (4) at a point ϕ ∈ M we need to take the pull-back of the complex (5) via ϕ, twist it by O_{C}(−1) and compute the second differential of the hypercohomology spectral sequence. The following Lemma describes this differential in general situation.

Lemma 2. Consider a complex K^{•} = (F ⟶^{f} A ⊗ O_{C} ⟶^{g} G) on C, where A is a vector space and

\begin{aligned}f & ∈ & Hom(F, A ⊗ O_{C}) & = & A ⊗ H^{0}(C, F^{∗}), \\ g & ∈ & Hom(A ⊗ O_{C}, G) & = & A^{∗} ⊗ H^{0}(C, G). \end{aligned}

Consider

D = tr(f ⊗ g) ∈ H^{0}(C, F^{∗}) ⊗ H^{0}(C, G) = H^{0}(C × C, F^{∗} ⊠ G),

where tr : A ⊗ A^{∗} → ℂ is the trace homomorphism. Then 1) The restriction of D to the diagonal Δ ⊂ C × C vanishes, hence D = D̃Δ for some

D̃ ∈ H^{0}(C × C, (F^{∗} ⊠ G)(−Δ)) = H^{0}(C, F^{∗}(−1)) ⊗ H^{0}(C, G(−1)) =

= H^{1}(C, F(−1))^{∗} ⊗ H^{0}(C, G(−1)).

2) The second differential d_{2} : H^{1}(C, F(−1)) → H^{0}(C, G(−1)) of the hypercohomology spectral sequence of K^{•} ⊗ O_{C}(−1) is induced by the section D̃.

Proof. The first statement is evident. To prove the second statement consider the following commutative diagram on C × C

F(−1) ⊠ O_{C}  ⟶^{f(−1)⊠1} A ⊗ O_{C×C}(−1, 0) ⟶^{(1⊠g)|_{Δ}} (O_{C}(−1) ⊠ G)|_{Δ}
D̃↓  ^{1⊠g}↓  ∥
O_{C}(−2) ⊠ G(−1) ⟶^{Δ} O_{C}(−1) ⊠ G ⟶^{|_{Δ}} (O_{C}(−1) ⊠ G)|_{Δ}

Both rows are complexes with acyclic middle term, hence the second differentials of the hypercohomology spectral sequences commute with the maps induced on cohomology by the vertical arrows:

H^{1}(C × C, F(−1) ⊠ O_{C}) ⟶^{d_{2}} H^{0}(C × C, (O_{C}(−1) ⊠ G)|_{Δ})
D̃↓  ∥
H^{1}(C × C, O_{C}(−2) ⊠ G(−1)) ⟶^{d_{2}} H^{0}(C × C, (O_{C}(−1) ⊠ G)|_{Δ})

Now it remains to note that

\begin{aligned}H^{1}(C × C, F(−1) ⊠ O_{C}) & = & H^{1}(C, F(−1)), \\ H^{1}(C × C, O_{C}(−2) ⊠ G(−1)) & = & H^{0}(C, G(−1)), \\ H^{0}(C × C, (O_{C}(−1) ⊠ G)|_{Δ}) & = & H^{0}(C, G(−1)), \end{aligned}

and that the map H^{0}(C, G(−1)) → H^{0}(C, G(−1)) induced by the map d_{2} in the second row of the above diagram is identity.

3.6. Consider the pullback of (5) via ϕ ∈ M, and twist it by O_{C}(−1). We want to apply Lemma 2 to compute the (ω_{i}, ω_{j})-component of the second differential of the hypercohomology spectral sequence.

In notations of the Lemma we have

D_{ω_{i},ω_{j}}(z, w) = ∑ ξ^{k}ϕ_{ω_{i}}(z) ⊗ ξ_{k}ϕ_{ω_{j}}(w) − ∑ ω_{i}(h^{i})ϕ_{ω_{i}}(z) ⊗ ω_{j}(h_{i})ϕ_{ω_{j}}(w) =

= ∑ ξ^{k}ϕ_{ω_{i}}(z) ⊗ ξ_{k}ϕ_{ω_{j}}(w) − (ω_{i}, ω_{j})ϕ_{ω_{i}}(z) ⊗ ϕ_{ω_{j}}(w) ∈ V_{ω_{i}} ⊗ V_{ω_{j}}(z, w).

Lemma 3. The operator ∑ ξ^{k} ⊗ ξ_{k} − (ω_{i}, ω_{j}) acts as a scalar multiplication on every irreducible summand V_{λ} ⊂ V_{ω_{i}} ⊗ V_{ω_{j}}.

On V_{ω_{i}+ω_{j}} it acts as a multiplication by 0.

If ω_{i} = ω_{j}, then on V_{2ω_{i}−i′} ⊂ V_{ω_{i}} ⊗ V_{ω_{i}} it acts as a multiplication by (−2).

If i ≠ j, i·j ≠ 0 then on V_{ω_{i}+ω_{j}−i′−j′} ⊂ V_{ω_{i}} ⊗V_{ω_{j}} it acts as a multiplication by (i·j −2).

Proof. It is easy to check that ∑ ξ^{k} ⊗ ξ_{k} commutes with the natural action of 𝔤 on V_{ω_{i}} ⊗ V_{ω_{j}}. The first part of the Lemma follows. The rest of the Lemma can be checked by the straightforward computation of the action of ∑ ξ^{k} ⊗ ξ_{k} on the highest vectors of the corresponding subrepresentations.

3.7. If we want to compute the brackets of the coordinates (4) we are interested in the components of D_{ω_{i},ω_{j}}(z, w) in the weights

ω_{i} + ω_{j}, ω_{i} + ω_{j} − i′, ω_{i} + ω_{j} − j′, ω_{i} + ω_{j} − i′ − j′.  (6)

The following Lemma describes the corresponding weight components of the tensor product V_{ω_{i}} ⊗ V_{ω_{j}}.

Lemma 4. The embedding V_{ω_{i}+ω_{j}} ⊂ V_{ω_{i}} ⊗ V_{ω_{j}} induces an isomorphism in the weights (6) with the following two exceptions:

(1) (V_{ω_{i}} ⊗ V_{ω_{i}})^{2ω_{i}−i′} = V_{2ω_{i}}^{2ω_{i}−i′} ⊕ V_{2ω_{i}−i′}^{2ω_{i}−i′}; the G-projection to the second summand is given by the formula

a(v_{ω_{i}} ⊗ F_{i}v_{ω_{i}}) + b(F_{i}v_{ω_{i}} ⊗ v_{ω_{i}}) ↦ \frac{a − b}{2}(v_{ω_{i}} ⊗ F_{i}v_{ω_{i}} − F_{i}v_{ω_{i}} ⊗ v_{ω_{i}}).

(2) (V_{ω_{i}} ⊗V_{ω_{j}})^{ω_{i}+ω_{j}−i′−j′} = V_{ω_{i}+ω_{j}}^{ω_{i}+ω_{j}−i′−j′} ⊕V_{ω_{i}+ω_{j}−i′−j′}^{ω_{i}+ω_{j}−i′−j′} if i ≠ j and i·j ≠ 0; the G-projection to the second summand is given by the formula

a(v_{ω_{i}} ⊗ F_{i}F_{j}v_{ω_{j}}) + b(F_{i}v_{ω_{i}} ⊗ F_{j}v_{ω_{j}}) + c(F_{j}F_{i}v_{ω_{i}} ⊗ v_{ω_{j}}) ↦

↦ \frac{b − a − c}{i · j − 2}(v_{ω_{i}} ⊗ F_{i}F_{j}v_{ω_{j}} + (i · j)F_{i}v_{ω_{i}} ⊗ F_{j}v_{ω_{j}} + F_{j}F_{i}v_{ω_{i}} ⊗ v_{ω_{j}}).

Proof. Straightforward.

A NOTE ON THE SYMPLECTIC STRUCTURE ON THE SPACE OF G-MONOPOLES 9

3.8. Hence (see Lemma 3, Lemma 4) when λ is one of the weights (6) the λ-component D̃_{ω_{i},ω_{j}}^{λ}(z, w) of the polynomial D̃_{ω_{i},ω_{j}}(z, w) = \frac{D_{ω_{i},ω_{j}}(z, w)}{z − w} is zero with the following two exceptions

D̃_{ω_{i},ω_{i}}^{2ω_{i}−i′} = \frac{ϕ_{ω_{i}}^{ω_{i}}(z)ϕ_{ω_{i}}^{ω_{i}−i′}(w) − ϕ_{ω_{i}}^{ω_{i}−i′}(z)ϕ_{ω_{i}}^{ω_{i}}(w)}{z − w}(F_{i}v_{ω_{i}} ⊗ v_{ω_{i}} − v_{ω_{i}} ⊗ F_{i}v_{ω_{i}})  (7)

D̃_{ω_{i},ω_{j}}^{ω_{i}+ω_{j}−i′−j′} = \frac{ϕ_{ω_{i}}^{ω_{i}−i′}(z)ϕ_{ω_{j}}^{ω_{j}−j′}(w) − ϕ_{ω_{i}}^{ω_{i}}(z)ϕ_{ω_{j}}^{ω_{j}−i′−j′}(w) − ϕ_{ω_{i}}^{ω_{i}−i′−j′}(z)ϕ_{ω_{j}}^{ω_{j}}(w)}{z − w} ·

· (v_{ω_{i}} ⊗ F_{i}F_{j}v_{ω_{j}} + (i · j)F_{i}v_{ω_{i}} ⊗ F_{j}v_{ω_{j}} + F_{j}F_{i}v_{ω_{i}} ⊗ v_{ω_{j}}) (8)

Note that the scalar multiplicators of the Lemma 3 canceled with the denominators of the Lemma 4. 3.9. Now we can compute the brackets. Proposition 2. We have

\begin{aligned}\{ & x_{ω_{i}}^{k} & , & x_{ω_{j}}^{l} & \} & = & 0; \\ \{ & x_{ω_{i}}^{k} & , & y_{ω_{j}}^{l} & \} & = & δ_{kl}δ_{ij}y_{ω_{j}}^{l}; \\ \{ & y_{ω_{i}}^{k} & , & x_{ω_{j}}^{l} & \} & = & −δ_{kl}δ_{ij}y_{ω_{i}}^{k}; \\ \{ & y_{ω_{i}}^{k} & , & y_{ω_{j}}^{l} & \} & = & i · j \frac{y_{ω_{i}}^{k}y_{ω_{j}}^{l}}{x_{ω_{i}}^{k} − x_{ω_{j}}^{l}} , & if i ≠ j; \\ \{ & y_{ω_{i}}^{k} & , & y_{ω_{i}}^{l} & \} & = & 0. \end{aligned}  (9)

Proof. Note that if p ∈ V_{ω_{i}}(z) then

dy_{ω_{i}}^{k}(p) = ⟨f_{ω_{i}}^{i′−ω_{i}}, p(x_{ω_{i}}^{k})⟩ , dx_{ω_{i}}^{k}(p) = ⟨f_{ω_{i}}^{−ω_{i}}, \frac{p(x_{ω_{i}}^{k})}{(ϕ_{ω_{i}}^{ω_{i}})′(x_{ω_{i}}^{k})}⟩ ,

where ⟨•, •⟩ stands for the natural pairing. Note also that

ϕ_{ω_{i}}^{ω_{i}}(x_{ω_{i}}^{k}) = 0, ϕ_{ω_{i}}^{ω_{i}−i′}(x_{ω_{i}}^{k}) = y_{ω_{i}}^{k}

by definition and

⟨f_{ω_{i}}^{i′−ω_{i}}, F_{i}v_{ω_{i}}⟩ = ⟨E_{i}f_{ω_{i}}^{−ω_{i}}, F_{i}v_{ω_{i}}⟩ = −⟨f_{ω_{i}}^{−ω_{i}}, E_{i}F_{i}v_{ω_{i}}⟩ = −1.

Now the Proposition follows from the Lemma 2 and from the formulas of 3.8.

Remark 2. The matrix of the bivector field P^{X} in the coordinates (x_{ω}^{k}, y_{ω}^{k}) looks as follows

0  diag(y_{ω}^{k})
− diag(y_{ω}^{k})  ∗

Since on the open set U this matrix is evidently nondegenerate it follows that the functions (x_{ω}^{k}, y_{ω}^{k}) indeed form an étale coordinate system.

3.10. Now we can prove Theorem 1.

Proof of the Theorem 1. The reduction to the case J = ∅ has been done in 2.2. The latter case is straightforward by the virtue of Proposition 2.

Corollary 1. The map P^{X} provides the space M_{b}(X, α) with a holomorphic symplectic structure.

Proof. Since P^{X} gives a Poisson structure it suffices to check that P^{X} is nondegenrate at any point. To this end recall that the hypercohomology spectral sequence of a complex K^{•} converges to H^{•}(X, K^{•}). Since the only nontrivial cohomology of the complex (1) is 𝔥_{X} in degree zero, the complex (2) is quasiisomorphic to (ϕ^{∗}𝔥) ⊗ O_{C}(−1) in degree zero, hence the hypercohomology sequence of the complex (2) converges to zero, hence P_{ϕ}^{X} is an isomorphism.

Remark 3. One can easily write down the corresponding symplectic form in the coordinates (4):

∑ \frac{dy_{ω}^{k} ∧ dx_{ω}^{k}}{y_{ω}^{k}} + \frac{1}{2} ∑_{i≠j} i · j \frac{dx_{ω_{i}}^{k} ∧ dx_{ω_{j}}^{l}}{x_{ω_{i}}^{k} − x_{ω_{j}}^{l}}

References

[1] M.F.Atiyah, N.Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Princeton University Press (1988).

[2] B.Feigin, A.Odesskii, Vector bundles on Elliptic Curves and Sklyanin Algebras, to appear in Proceedings of IUM.

[3] M.Finkelberg, I.Mirković, Semiinfinite Flags. I. Case of global curve ℙ^{1}, Preprint alg-geom/9707010.

[4] S.Jarvis, Euclidean monopoles and rational maps, to appear in Proc. Lond. Math. Soc.

Independent Moscow University, 11 Bolshoj Vlasjevskij pereulok, Moscow 121002 Russia

  E-mail address: fnklberg@mccme.ru

Independent Moscow University, 11 Bolshoj Vlasjevskij pereulok, Moscow 121002 Russia

  E-mail address: sasha@kuznetsov.mccme.ru kuznetsov@mpim-bonn.mpg.de

Dept. of Mathematics and Statistics, University of Massachusetts at Amherst, Amherst MA 01003-4515, USA

  E-mail address: mirkovic@math.umass.edu